Factors with only two variables in factor analysis

I am running a factor analysis and have a couple of questions.

I have 10 variables, all of them come from a survey, with each answer is in the scale of 1 to 7. I have calculated a correlation matrix between the 10 variables, and looked for correlations of 0.5 or higher. These made me notice of 4 groups of correlated variables, two groups of 3 variables and two groups of 2 variables.

Then I ran the factor analysis. According to the scree plot, I should take 4 factors. According to the eigenvalues, the 4th eigenvalue is 1.006, so it's slightly on the line, it is larger than 1, but not by much...In addition, after 4 factor the cumulative variance is 82%. After 3 it is only 72%.

My question is, should I choose 4 factors or 3 ? Is it OK to have a factor with only two variables that construct it ? I know that a single variable factor is something we don't want, just like we don't want a sample-specific factors. But what is the rule regarding two variables in a factor ? I looked on the net, and found one document saying the minimum is 3. I didn't see any mathematical explanation and I am not convinced. What can you recommend me ?

• Scree and the K1 method seem suggest that you could retain the 4th factor. Maybe clarifying the ultimate goal of your analysis and setting a performance metric for your modeling could help deciding if you ultimately should include the 4th factor. Commented Apr 23, 2014 at 13:31
• A rule of thumb motivated by some properties of FA is "3+ variableds on a factor". So I would recommed you 3 factors. But that is not a law. If you see 4 factors reproduce correlations much better and the factors are better interpretable, choose 4. Then I'd recommend not to interpret the factor with 2 items. Try to get more items to support it (if that is possible) and redo FA. Commented Jan 11, 2017 at 17:16
• Given observations (1 observation = 1 vector of variable values) and you want to find the relationships between the 10 variables, I suggest Greg Ver Steeg CorEx method. You can start with the linear sieve which is fast. isi.edu/projects/gregv/correlation_explanation The python package(s) also offer nice visualization of dependencies. Commented Jan 5, 2018 at 4:03
• Also I caution you against relying on the "eigenvalue >1" rule. See any source using the phrase 'tom swift's electric factor analysis machine'. Commented Apr 11, 2018 at 14:05

With factor analysis, we have a lot of guidelines and no strict rules. Eigenvalues over 1, scree plot, number of variables to a factor, amount of variance explained and others are all guidelines.

My suggestion is that you look at the two solutions and see which one makes more sense. Are the three factors useful constructs? When you go from 3 to 4, do you more-or-less keep the original 3 and add a 4th, or does the whole solution change? If the former, then is the 4th factor useful? If the latter, then is the new solution more sensible or less sensible?

You should run Monte Carlo PCA for Parallel Analysis, you will get simulated eigen values, compare them with the real data eigen values you got, if the simulated eigen value is less than the real data eigen value then retain otherwise leave. For downloading Monte Carlo PCA for Parallel Analysis.